Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 5 Number Sense
Problem
Hint
1) What is the time exactly half way between 9:37 pm and 2:43 am? 1. Compute the number of minutes from 9:37 to midnight:
______
2. Compute the number of minutes from midnight to 2:43 am = _____
3. Add these and divide by 2: _______
4. If this is greater than #1 above, subtract that from it and convert back to hours and minutes _____
5. If this is not greater than #1 above, add it to #2 and convert back to hours and minutes = _____.
6. The half way time is _____.

2) A bowling ball and a bag together cost $88. The ball costs three times as much as the bag. How much does the bowling ball cost? 		This is an algebra problem!
Let B = cost of the bag.
Then:
3B + B = $88.
Solve this equation. B (the bag) = _____
Cost of the bowling ball = $_______.

3) Kyle baked at least two dozen cookies but fewer than 60 cookies. If he divides the cookies evenly among 7 plates, 4 cookies are left over. If he divides the cookies evenly among 6 plates, 5 cookies are left over. How many cookies did Kyle bake? 		Make the following table:
Cookies
per plate		times 7 +4 times 6 +5
43225
5  
6  
7  
8  
9  
Fill out this table until you find an entry in the "times 7+4" column that matches another entry in the "times 6+5" column. That entry is the number of cookies Kyle baked = _____.

Problem
Hint

4) The sum of two numbers is 16 and their product is 48. What is the ratio of the two numbers when expressed as an improper fraction in lowest terms ? 		1. Make a table of numbers whose product is 48:
  1. 8 x 6 = 48
  2. . . .
2. Find two numbers in the table above whose sum is 16: ___ and ___
3. Divide the larger one by the smaller one: = __________
4. Reduce to lowest terms: = _________

5) Four friends have their birthdays in the same month. This year the squares containing their birthdates on the calendar touch each other such that they make a larger square. Two of their birthdates are multiples of six. The horizontal sums and the vertical sums are prime. What is the sum of the four birthdates?		 OK, a tough problem!
1. The place to start is with the horizontal sums of 2 days of the month that add to a prime number. Here they are:

(1+2) (2+3) (3+4) (5+6) (6+7) (8+9) (9+10) (11+12)
(14+15) (15+16) (18+19) (20+21) (21+22) (23+24)
(26+27) (28+29) (29+30) (30+31)
2. Now, with this list, the dates under the pairs are these dates +7. For example, (8+9) goes under (1+2). Write out the pairs (one pair above the other) that fit this requirement:


3. Select out of this list those that add vertically to primes:



4. Finally, Select out of this list those in which 2 of the 4 dates are multiples of 6:


5. The dates that satisfy this last requirement are:

6. Sum the 4 dates = _____